In the electricity market, the objective of users is to maximize their individual welfare by adjusting power consumption and managing renewable energy generation and energy storage systems. It is assumed that the electricity supplier can fully meet the power demand of all users.

3.1.1. Utility Function

The utility function is a concept from microeconomics that establishes a mathematical relationship between the quantity of commodity consumption and consumer subjective satisfaction. In power systems, user energy consumption behavior can be characterized by a quadratic utility function. This function adheres to the law of diminishing marginal utility and exhibits concavity, where marginal utility slows down as consumption increases, eventually stabilizing at a saturation point. While various forms such as logistic and logarithmic functions exist, this paper adopts the quadratic utility function, as it effectively captures the principle of diminishing marginal utility. This choice is widely recognized in power system economics for accurately representing rational consumer satisfaction while maintaining computational tractability for optimization [17]. This paper adopts the quadratic utility function. For a user n located at node i ( n ∈ N i M G , i ∈ N ) during time slot t , the utility function U i n , t M G is defined as a piecewise function:

where P i n , t L is the active power consumption of user n at node i in time slot t . ω i n , t and α are non-negative preset parameters. ω i n , t represents the basic demand intensity, and α characterizes the rate of diminishing marginal utility. Assuming users operate within the non-saturated region, the load is subject to lower and upper limit constraints:

where P i n L , min and P i n L , max are the minimum and maximum power consumption limits for user n at node i .

3.1.2. Distributed Renewable Energy

In power systems, the distributed renewable energy equipment equipped by users is typically small-scale, such as rooftop photovoltaics (PV) or small wind power generation devices, primarily used to meet the user’s own electricity demand. Given their limited installed capacity, the generated energy is usually not sold back to the grid and is limited to self-consumption.

For small-scale renewable energy output, a deterministic generation curve is adopted, fitted through historical data or generated from typical scenarios, to maintain model accuracy while ensuring computational efficiency. Assuming the PV generation quantity P i n , t R E S is an exogenous parameter satisfying the constraint: P i n , t R E S ∈ [ 0 , P i n , t R E S , max ] , where P i n , t R E S , max is the maximum PV generation capability in time slot t . Assuming the generation cost of renewable energy is negligible, the maintenance and depreciation loss costs of the renewable energy generation system should still be accounted for, modeled using a quadratic function [34]:

where γ i n R E S and δ i n R E S are non-negative preset parameters.

3.1.3. Energy Storage System

Energy Storage Systems (ESS) play a key role in power management by smoothing electricity costs through flexible charging and discharging strategies and enhancing the stability and reliability of power supply. Their dynamic behavior and related characteristics are modeled as follows:

1)Charging and Discharging Dynamics

The capacity state of the ESS changes over time. Its update equation and related constraints are:

where P i n , t E S S is the charging or discharging power of the storage device in time slot t . When P i n , t E S S > 0 , it indicates charging; when P i n , t E S S < 0 , it indicates discharging. E i n , t represents the energy storage capacity state in time slot t , reflecting the remaining power of the storage device. E i n m i n and E i n m a x are the minimum and maximum capacities of the ESS. P i n E S S , max is the maximum power limit for charging (or discharging), which is a preset parameter ensuring that charging and discharging operations are performed within the power range the device can withstand. The equation ∑ t = 1 T Δ t P i n , t E S S = 0 represents the periodic constraint for storage scheduling.

2)Charging and Discharging Costs

The operating cost of the ESS mainly stems from equipment depreciation and maintenance, which is related to the amount of charging and discharging energy. It is modeled using the following function [22]:

where γ i n E S S and δ i n E S S are non-negative preset parameters.

3.1.4. Total Welfare of Demand Side

Considering users equipped with distributed photovoltaics and energy storage, their total welfare consists of electricity utility minus electricity purchase costs, renewable energy equipment maintenance costs, and energy storage operating costs. The net power purchased by the user from the grid is:

where P i n , t M G is the power purchased by the user from the grid in time slot t , representing the interaction power. λ i , t is the real-time electricity price at node i in time slot t . To prevent market disorder caused by excessively low or high electricity prices, upper and lower limit constraints are applied to the electricity price:

where λ i , t m i n and λ i , t m a x are preset parameters.

The objective function of the demand side is to maximize total welfare, which can be formulated as:

For ease of subsequent solution, this is transformed into:

where the first term is negative utility, the second and third terms are photovoltaic maintenance costs and energy storage operating costs respectively, and the fourth term is the electricity purchase cost.

In the power market scheduling system, after collecting and integrating all users’ consumption data, the system’s objective is to maximize the overall profit of the generation side by optimizing electricity prices and power supply dispatch schemes. To achieve this, the system needs to ensure that the total generation complies with distribution network power flow constraints while satisfying total user demand, guaranteeing the feasibility of power transmission and the safe and stable operation of the grid. Furthermore, it is necessary to coordinate pricing strategies and generation plans to maximize profits based on supply-demand balance and improve system operational efficiency.

3.2.1. Power Transmission Loss Cost

The DistFlow power flow model is widely used in radial distribution network analysis [35]. Based on branch power, this model is more suitable for radial distribution systems compared to traditional node-power-based methods. In this model, power losses must be accurately calculated based on node voltage magnitudes, line resistance, and reactance. For the line l i j connecting node i and node j , denote i j as the corresponding line l i j . The basic power flow equations are as follows [36]:

where U i , t and U j , t are the voltage magnitudes of the corresponding nodes in time slot t . P i j , t and Q i j , t are the active and reactive power flowing from node i to node j on line i j in time slot t . P j , t i n and Q j , t i n are the active and reactive power injections at node j , respectively. ∑ k : j → k P j k , t represents the sum of active power of all downstream branches starting from node j in time slot t , and ∑ k : j → k Q j k , t represents the sum of reactive power of all downstream branches starting from node j in time slot t . x i j and r i j represent the reactance and resistance of line i j . I i j , t represents the current of line i j in time slot t . The system must satisfy the following operational constraints:

where U i min and U i max are the lower and upper limits of the voltage magnitude at node i , and I i j max is the maximum current limit for line i j , all of which are preset parameters.

Since the current-power relationship in the model, i.e., U i , t 2 I i j , t 2 = P i j , t 2 + Q i j , t 2 , is non-linear, it leads to non-convexity in the optimization problem, increasing the difficulty of solution. To obtain a convex optimization form, Second-Order Cone Programming (SOCP) is used to relax the aforementioned relationship. The relaxed problem is a convex problem, and using existing convex optimization algorithms guarantees finding the global optimal solution [37]. This SOCP relaxation is highly effective for radial distribution systems as it preserves the physical characteristics of the power flow while transforming the non-convex constraints into a computationally tractable form. Under most operational conditions, this relaxation is proven to be exact, ensuring that the optimal solution of the relaxed model remains physically feasible for the original power network. Let V i , t = U i , t 2 and ℐ i j , t = I i j , t 2 . The relationship is relaxed to:

In AC systems, power loss caused by resistance follows Joule’s law and is proportional to the square of the current. The power loss of line i j in time slot t is:

Then the transmission loss cost of line i j in time slot t is:

where λ i j , t L o s s represents the cost coefficient for losses of line i j at time t .

3.2.2. Conventional Generation Cost

The cost of traditional thermal power generation units is commonly represented by a quadratic function. Let P i g , t G and Q i g , t G denote the active and reactive power output of the g -th generator at node i in time slot t , respectively. The following operational constraints must be satisfied:

where P i g G , min , P i g G , max , Q i g G , min , and Q i g G , max represent the minimum and maximum active and reactive power outputs of the g -th generation unit at node i , respectively. Δ P i g G , max and Δ Q i g G , max represent the maximum ramp rates in adjacent time slots.

The generation cost is only related to active power output, and reactive power output is not included in the main cost function. That is, the generation cost function of unit g in time slot t can be expressed as:

where a i g , b i g , and c i g are the generation cost function coefficients for unit g , all of which are preset parameters.

3.2.3. Main Grid Power Purchase Cost

Distributed generation resources within the distribution network can satisfy local load demands to a certain extent, but their marginal generation costs typically rise with increased output. Therefore, under the condition of satisfying overall economic optimality, the PMSC can choose to purchase a portion of electricity from the upper-level main grid to better utilize the cost advantages of external resources and achieve global economic dispatch. Assuming node 1 is the interconnection point between the distribution network and the main grid, the cost of purchasing electricity from the main grid can be expressed as:

where P 1 , t E X is the interaction power between the supply-side distribution network and the upper-level main grid, and λ t E X represents the electricity price provided by the upper-level main grid to the supply-side distribution network, which is a preset parameter.

3.2.4. Supply Side Welfare

The total welfare of the supply side is reflected as total electricity sales revenue minus total generation costs, transmission loss costs, and power purchase costs from the upper-level main grid. The congestion costs have already been reflected through the differences in nodal prices. To ensure the safe and stable operation of the power grid, real-time supply-demand balance and power flow constraints must be satisfied, as follows:

where P i j min , P i j max , Q i j min , Q i j max are the minimum and maximum active and reactive power allowed for line $ij$, which are all preset parameters. P i , t i n and Q i , t i n are the active and reactive power injections at node i , respectively. Q i , t L is the total reactive power load at node i , and Q i n , t M G is the net reactive load of user i n . To simplify the model, Q i , t L is defined as a preset parameter, and ζ is a preset parameter representing the proportional relationship between net reactive power and net active power.

The total sales revenue and the corresponding price constraints are in the following form:

The overall optimization objective of the supply side is to maximize total welfare:

where the first term is the total sales revenue, the second term is the total generation cost, the third term is the total loss cost, and the last term is the transaction cost with the main grid.

The upper-level decision maker is the Power Market Scheduling Center (PMSC), which involves multiple nodes and generation units. It needs to formulate real-time nodal prices and supply plans to maximize its profit. This problem can be formulated as a non-linear programming model subject to various physical constraints (such as power balance and line capacity). The objective function comprehensively considers generation costs, electricity sales revenue, and network loss costs. Based on the system constraints and objective function described in the previous sections, the upper-level problem is formalized as:

where the first term W is the total electricity sales revenue defined in Equation (19), the traditional thermal generation cost C i g , t G is defined in Equation (16), the distribution network loss cost C i j , t L o s s is defined in Equation (14), and the cost of purchasing power from the upper-level main grid C T is defined in Equation (17).

The lower-level decision makers are the user groups, who maximize their welfare by adjusting electricity loads and energy storage operations. This problem is modeled as a convex quadratic programming model. Users respond to price signals by optimizing their consumption behavior and storage strategies. Based on the user load response constraints, the lower-level problem is formalized as:

where the user utility U i n , t M G is defined in Equation (1), the renewable energy equipment depreciation and maintenance cost C i n , t R E S is defined in Equation (3), the energy storage charging/discharging cost C i n , t E S S is defined in Equation (5), and the electricity purchase cost C i n , t E X is defined in Equation (8). In the lower-level problem, the depreciation and maintenance cost functions for renewable energy and storage devices are convex, and the utility function is concave; therefore, the lower-level problem (22) is a convex optimization problem.

To address the typical “upper-level maximization, lower-level minimization” structure of bi-level optimization, and to improve solution efficiency, KKT optimality conditions and duality theory are utilized to transform the original problem into an equivalent single-level mathematical programming problem. The specific steps are as follows:

4.3.1. Construction of Lagrangian Function for Lower-Level Problem

For the clarity of the solution process, the subsequent transformation targets a specific user under node i (subscripts regarding the user index are implicitly handled in the standard notation i n ). The Lagrangian function is constructed as:

4.3.2. KKT Conditions

Since the lower-level problem is a convex quadratic programming problem and satisfies Slater’s constraint qualification, its optimal solution must satisfy the KKT necessary and sufficient conditions.

4.3.3. Linearization of Non-Linear Constraints

For the complementary slackness terms in the KKT conditions in the form of x ⋅ y = 0 , ( x , y ≥ 0 ) , the Big-M method is employed for linearization. A binary variable θ ∈ { 0 , 1 } and a sufficiently large positive real number M are introduced to construct the equivalent constraints:

When θ = 1 , 0 ≤ x ≤ M and y = 0 ; when θ = 0 , x = 0 and 0 ≤ y ≤ M . Thus, the linearization transformation of the non-linear terms is achieved.

4.3.4. Application of Strong Duality Theorem

Based on the strong duality theorem, the optimal solution of the lower-level problem can be equivalently described by its dual problem. The equivalent expression for the optimal solution of the lower-level problem derived from the strong duality theorem is:

In summary, after substituting the KKT conditions into the upper-level problem, the master-slave game model can be transformed into the following Mixed-Integer Linear Programming (MILP) problem:

This problem belongs to the class of mixed-integer linear programming problems and can be efficiently solved using commercial solvers.

The structure of the simulation system is shown in Figure 1. The supply side consists of two parts: first, small-scale thermal power generating units deployed at nodes 6, 8, and 13 of the distribution network; second, electricity trading with the upper-level main grid via the connection at node 1. The demand side includes three independent microgrids located at nodes 8, 18, and 33, with each microgrid containing three residential users. Each user is equipped with photovoltaic (PV) generation equipment, wind power generation equipment, and a home energy storage system. The scheduling period is set to one day, divided into 24 equal time slots ( t = 1 , 2 , ⋯ , 24 ). The cost function of the generator sets is defined by Equation (16), and the relevant parameters are listed in Table 1.

Table 1. Relevant parameters of generator sets.

The user electricity utility function is defined as Equation (1), where ω ∈ [ 1 , 5 ] takes random values and α = 0.1 . The parameter ω for different users takes different values in different time slots. The total capacity of solar PV and wind power generation equipment installed by each user is 10 kW, and the capacity of the energy storage equipment is 10 kWh. Figure 2 depicts the output of solar PV and wind power generation within a day based on statistical characteristics. Although the marginal cost of renewable energy generation is zero, equipment maintenance and depreciation costs must still be considered. The parameters are set as γ i n R E S = γ i n E S S = 0 and δ i n R E S = δ i n E S S = 0.01 . The wholesale electricity price released by the main grid is known, and its price curve is visible in Figure 3. The real-time electricity price released by the PMSC is required to fluctuate within a certain range: the lower limit is set at 0.8 times the main grid benchmark price, and the upper limit is set at 3 times. Additionally, ζ = 0.5 , assuming that the net reactive power is 0.5 times the net active power.

Figure 2. Solar and wind power output of user 3.

Based on the parameter settings mentioned above, Figure 3 illustrates the optimal electricity prices for each microgrid under the proposed real-time locational marginal pricing strategy, alongside the benchmark price of the upper-level main grid. Figure 4 presents the optimal dispatch scheme for the system’s total consumption, total supply, renewable energy, and energy storage. As depicted in Figure 4, the total consumption by users exceeds the total supply, indicating that users utilized power generated from their own renewable energy sources or discharged from energy storage devices. In fact, by adhering to constraint (6), the

Figure 3. Time-series comparison of nodal real-time electricity prices in the microgrid and the benchmark main grid prices.

Figure 4. Time-series variation of system power balance.

balance between the supply side and the demand side is strictly maintained. Taking User 3 as an example, Figure 5 displays the optimal energy dispatch scheme on the demand side over the operation cycle, covering electricity consumption, renewable energy output, and charging/discharging energy. Figure 6 illustrates the optimal power supply dispatch on the supply side.

Based on Figures 3-6, it can be concluded that throughout the scheduling cycle, electricity price fluctuations are correlated with user consumption preferences, renewable energy production, the benchmark price of the upper-level main grid, and the total cost of generation units. During periods with lower energy consumption and lower main grid prices, such as time slots 1 - 6 and 22 - 24, or periods with higher renewable energy output, such as time slots 10 - 14, electricity prices are relatively low. Conversely, during periods of high energy consumption and high main grid prices, such as time slots 9 - 10 and 19 - 21, electricity prices are relatively high, indicating that users actively participate in electricity market regulation. Furthermore, differences in the amplitude of price fluctuations exist among different microgrids, which primarily stem from three factors: first, there are differences in electricity consumption preferences among users in different microgrids; second, generation costs vary for units at different locations in the distribution system, allowing nodes located near low-cost power sources to prioritize the utilization of more economical power resources; third, nodal prices are also influenced by line transmission distances and network losses. In the proposed

Figure 5. Electricity dispatch curve for user 3 at the microgrid node.

Figure 6. Optimal supply-side dispatch scheme.

model, the LMP is decomposed into marginal generation cost and marginal loss cost. By solving the dual variables of the power flow constraints (10)-(12), the mechanism specifically quantifies the additional cost required to supply an incremental unit of load at a specific node while accounting for the corresponding increase in system-wide ohmic losses, which is then directly integrated into the final nodal price.

To verify the effectiveness and robustness of our model, operational results are compared under two different renewable energy configuration ratios. As described in Section 4.1, in Scenario (a), users are equipped with 5 kW photovoltaic (PV) and wind power generation equipment, whereas in Scenario (b), the installed capacity of both types of renewable energy equipment is increased to 10 kW. The simulation assumptions and parameter settings remain consistent across both scenarios. Figure 7 illustrates the PV and wind power output over a single day under Scenario (b).

To conduct an in-depth analysis of the impact of increased renewable energy penetration on grid operation and the interaction mechanism between supply and demand, a comparative analysis is performed from the perspective of multiple indicators. Figure 8 displays the optimized energy dispatch scheme for User 3 in Scenario (b). Compared with Scenario (a), the user’s optimal consumption plan

Figure 7. Solar and wind power output of user 3 in Scenario B.

Figure 8. Electricity dispatch curve for user 3 at the microgrid node in scenario B.

remains substantially consistent, primarily due to the similarity in load characteristics and daily consumption patterns between the two scenarios. However, as the installed capacity of renewable energy increases, the proportion of electricity purchased from the grid significantly decreases, indicating a greater tendency for users to utilize local renewable energy to satisfy their own power demands. Based on the analysis of the supply-demand interaction mechanism, the PMSC adopts a strategy of reducing generation to achieve supply-demand balance in this situation. As shown in Figure 9, the optimal supply-side dispatch scheme in this scenario corroborates this observation.

Figure 9. Optimal supply-side dispatch scheme under scenario B.

Particularly during daytime periods with abundant solar energy, the output of traditional generators exhibits a significant decline. This indicates that in power systems with a high proportion of renewable energy, characteristics such as the intermittency and volatility of renewable energy output exert a more pronounced inhibitory effect on the generation schedules of thermal units. Additionally, Figure 10 compares the social welfare levels under the two scenarios. The results show that social welfare in Scenario (b) is higher than that in Scenario (a), primarily attributed to the lower generation costs of renewable energy. This result not only confirms the significant value of renewable energy in enhancing the economic efficiency and sustainability of power systems but also validates the

Figure 10. Social welfare in Scenario A vs. Scenario B.

effectiveness of the proposed pricing mechanism in environments with high renewable energy integration. The mechanism helps incentivize the accommodation and development of renewable energy, thereby reducing dependence on traditional fossil fuels and promoting the reduction of carbon emissions in system operation.

To evaluate the performance of the proposed pricing mechanism, this subsection compares the proposed real-time locational marginal pricing mechanism with the traditional fixed pricing mechanism [17] under consistent simulation conditions.

Figure 11 illustrates the hourly variations in social welfare under the two mechanisms throughout the full operation cycle. The results indicate that the real-time locational marginal price achieved a total social welfare of 12039.97 CNY, representing an increase of 9.45% compared to 10999.96 CNY under the fixed pricing mechanism. This demonstrates that real-time locational pricing possesses significant advantages in enhancing the overall economic efficiency of the system.